The reliability of bolted assemblies is mainly a function of the level of the initial clamping load and the stability of the clamping load over the life of the joint. The initial level of clamping load is determined by the bolt preload achieved during the initial tightening of the bolted joint, which is often estimated based on torque level. However, the torque-tension relationship of a threaded bolt is highly sensitive to the friction variations between threads and under the turning bolt head or nut. Even moderate friction variations cause large scatter in the torque-tension correlation, which may compromise the reliability of the bolted joints for which the clamping force is estimated based solely on the torque level.
For critical applications, the bolt preload may be determined more accurately by measuring the bolt elongation caused by tightening. In contrast with the torque-tension relationship, friction plays no role in the correlation between bolt stretch and bolt tension. Bolt tension and the resulting clamping force in a bolted joint are directly proportional to the bolt elongation. So, the tightening process may be controlled by monitoring the bolt elongation. Similarly, the residual tension in a bolt remains directly proportional to the residual bolt elongation.
In the elastic range, the bolt tension F is given byF=Kb*Δl   (1)where Kb is the spring rate of the bolt (lb/in, N/mm), and Δl is the bolt elongation. This relationship is depicted in FIG. 1.
As is known, the spring rate of the bolt, Kb, can be determined experimentally through a load-elongation test of the same grip length of the bolt, or by developing an analytical model that provides the bolt spring rate. Obviously, the bolt elongation that corresponds to a desired preload level depends on the grip length of the bolt. Hence, bolts with shorter grip length will experience smaller elongations, which must be measured precisely in order to reduce the percentage error in the elongation measurement. Sheet metal applications provide examples for short grip lengths. In such applications, the bolt elongation may be very small, and hence this requires high precision measurements that ultrasonic technology may offer.
With reference to FIG. 2, the main principle in using ultrasonics to measure bolt length or bolt elongation is to monitor the round trip time for a longitudinal wave 200 to travel through bolt 202 and back to a transducer 204 that is mounted on an end of the bolt 204.
Ultrasonics have been used to control bolt tightening. One such technique is discussed in Nassar et al., “Controlling the turn of the screw,” Mechanical engineering magazine, vol. 113, no. 9, September 1991, pp. 52-56 (which is incorporated by reference herein in its entirety) However, this techniques involves monitoring and controlling the tightening process by using a constant, stress-independent, wave speed in order to use change in the round trip time to obtain bolt elongation. This, however, does not take into account the fact that the wave speed changes as the bolt is elongated during tightening. To compensate for this wave speed variation, this technique uses a correction factor called stress factor (“SF”), which is commonly obtained by mechanical calibration using gage blocks in a tension elongation test.
Ultrasonic wave speed is stress and elongation dependent. The speed of sound in a material is affected by the stress field. Higher stress impedes the transmission of ultrasonic waves in the bolt, making the round trip time of the wave longer. This makes the change in the bolt length appear to be larger than the actual elongation. The temperature dependence of the ultrasonic speed increases or decreases depending on whether the stress is applied parallel or perpendicular to the direction of the wave propagation.
For longitudinal waves through the bolt, only the axial stress level will cause changes in the wave speed. Stress due to shear loading or torsional stresses does not affect the sound velocity along the length of the bolt. The change in the wave speed is linear with respect to the stress level. It increases or decreases according to whether the stress is applied parallel to or perpendicular to wave propagation respectively. For a longitudinal ultrasonic wave propagating parallel to the direction of the applied axial stress, the governing equation, as discussed in “Measurement of Residual Stress Using the Temperature Dependence of Ultrasonic Velocity,” K. Salama, G. C. Barber, and N. Chandrasekaran, Proceedings of IEEE Ultrasonic Symposium, 1982, p. 877, is:
                                          ⅆ            v                                ⅆ            σ                          =                              -                          (                                                2                  ⁢                  l                                +                λ                +                                                      (                                                                  λ                        +                        μ                                            μ                                        )                                    ⁢                                      (                                                                  4                        ⁢                        m                                            +                                              4                        ⁢                                                                                                  ⁢                        λ                                            +                                              10                        ⁢                                                                                                  ⁢                        μ                                                              )                                                              )                                            2            ⁢            v            ⁢                                                  ⁢            ρ            ⁢                                                  ⁢                          (                                                3                  ⁢                                                                          ⁢                  λ                                +                                  2                  ⁢                                                                          ⁢                  μ                                            )                                                          (        2        )            Where λ and μ are lame or second-order elastic constants; l and m are Murnaghan's third-order elastic constants; ρ is density, ν is wave speed and σ is the compressive stress.
Due to the fact that tightened bolts are subjected to positive tensile stress, equation (2) is rewritten for bolts as follows:
                                          ⅆ            v                                ⅆ            σ                          =                              (                                          2                ⁢                l                            +              λ              +                                                (                                                            λ                      +                      μ                                        μ                                    )                                ⁢                                  (                                                            4                      ⁢                      m                                        +                                          4                      ⁢                                                                                          ⁢                      λ                                        +                                          10                      ⁢                                                                                          ⁢                      μ                                                        )                                                      )                                2            ⁢            ρ            ⁢                                                  ⁢            v            ⁢                                                  ⁢                          (                                                3                  ⁢                                                                          ⁢                  λ                                +                                  2                  ⁢                                                                          ⁢                  μ                                            )                                                          (        3        )            
Equation (3) may be integrated to yield:
                                                        v              2                        -                          v              0              2                                2                =                                            (                                                2                  ⁢                  l                                +                λ                +                                                      (                                                                  λ                        +                        μ                                            μ                                        )                                    ⁢                                      (                                                                  4                        ⁢                        m                                            +                                              4                        ⁢                                                                                                  ⁢                        λ                                            +                                              10                        ⁢                                                                                                  ⁢                        μ                                                              )                                                              )                        ⁢                                                  ⁢            σ                                2            ⁢                                                  ⁢            ρ            ⁢                                                  ⁢                          (                                                3                  ⁢                                                                          ⁢                  λ                                +                                  2                  ⁢                                                                          ⁢                  μ                                            )                                                          (        4        )            where ν is wave speed in stressed bolt and ν0 is zero stress wave speed.
In the elastic range, the axial stress σ may be expressed in terms of the axial elongation Δl as follows:
                                                        v              2                        -                          v              0              2                                2                =                                            (                                                2                  ⁢                  l                                +                λ                +                                                      (                                                                  λ                        +                        μ                                            μ                                        )                                    ⁢                                      (                                                                  4                        ⁢                        m                                            +                                              4                        ⁢                                                                                                  ⁢                        λ                                            +                                              10                        ⁢                                                                                                  ⁢                        μ                                                              )                                                              )                        ⁢                                                  ⁢                          E              ⁡                              (                                  Δ                  ⁢                                                                          ⁢                                      l                    /                    L                                                  )                                                          2            ⁢                                                  ⁢            ρ            ⁢                                                  ⁢                          (                                                3                  ⁢                                                                          ⁢                  λ                                +                                  2                  ⁢                                                                          ⁢                  μ                                            )                                                          (        5        )            where Δl/L is the axial strain of the bolt.
The wave speed is given in terms of bolt elongation and material properties by:
                    v        =                              [                                          v                0                2                            +                              [                                                                            E                      ⁡                                              (                                                  Δ                          ⁢                                                                                                          ⁢                          l                                                )                                                              [                                                                  2                        ⁢                        l                                            +                      λ                      +                                                                        (                                                                                    λ                              +                              μ                                                        μ                                                    )                                                ⁢                                                  (                                                                                    4                              ⁢                              m                                                        +                                                          4                              ⁢                                                                                                                          ⁢                              λ                                                        +                                                          10                              ⁢                                                                                                                          ⁢                              μ                                                                                )                                                                                      ]                                                        L                    ⁢                                                                                  ⁢                    ρ                    ⁢                                                                                  ⁢                                          (                                                                        3                          ⁢                                                                                                          ⁢                          λ                                                +                                                  2                          ⁢                                                                                                          ⁢                          μ                                                                    )                                                                      ]                                      ]                                1            /            2                                              (        6        )            
The wave speed after the bolt is stressed depends on initial speed of the longitudinal wave in the bolt, bolt elongation and the material properties. In equation (6), the material properties are constant except the density of the bolt material. The initial density ρ0 of the stressed segment of the bolt material is given by:
                              Initial          ⁢                                          ⁢          density          ⁢                                          ⁢                      ρ            0                          =                  M                      V            0                                              (        7        )            The density ρ of the stressed segment of the bolt is given by:
                    ρ        =                  M          V                                    (        8        )            The density change Δρ in the stressed segment of the bolt is given by:
                              Δ          ⁢                                          ⁢          ρ                =                              M            V                    -                      M                          V              0                                                          (        9        )            In the elastic range, the change in volume per unit volume is:
                                          Δ            ⁢                                                  ⁢            V                    V                =                              ɛ            X                    +                      ɛ            Y                    +                      ɛ            Z                                              (        10        )            
If the bolt is subjected to uniaxial stress, then stresses σY=σZ=0. The change in volume then becomes:
                                          Δ            ⁢                                                  ⁢            V                                V            0                          =                              (                          1              -                              2                ⁢                v                                      )                    ⁢                                    Δ              ⁢                                                          ⁢              l                        L                                              (        11        )            The change in density is then given by:
                              Δ          ⁢                                          ⁢          ρ                =                              M                                          V                0                            +                              Δ                ⁢                                                                  ⁢                V                                              -                      M                          V              0                                                          (        12        )            Using equations (11) and (12), the change in density is given by:
                              Δ          ⁢                                          ⁢          ρ                =                  -                                    M                              V                0                                      ⁡                          [                                                                    (                                          1                      -                                              2                        ⁢                        v                                                              )                                    ⁢                  Δ                  ⁢                                                                          ⁢                  l                                                  L                  +                                                            (                                              1                        -                                                  2                          ⁢                          v                                                                    )                                        ⁢                    Δ                    ⁢                                                                                  ⁢                    l                                                              ]                                                          (        13        )            where L is the initial length of the bolt, Δl is the elongation of the bolt, M is the mass of the stressed segment of the bolt, V is the volume of the stressed segment of the bolt, V0 is the initial volume, ΔV change in bolt volume due to bolt elongation and ν is Poisson's ratio.
FIG. 3 illustrates the non-dimensional change in the density of the bolt material according to equation (13). As can be seen from FIG. 3, in the elastic range, the density change is negligible.
Stress level in the bolt affects the temperature dependence of the wave speed. The effect of stress on the temperature dependence of longitudinal ultrasonic wave speed becomes much smaller, and opposite in sign, when the stress is applied parallel to the direction in which the waves are propagated.